An unconditionally stable numerical approach for solving a nonlinear distributed delay Sobolev model

TL;DR

This paper introduces an unconditionally stable, high-order accurate numerical method for nonlinear Sobolev models with distributed delay, combining interpolation for time derivatives and finite element methods for spatial derivatives.

Contribution

It presents a novel, simple, and efficient numerical approach with rigorous stability and error analysis for delay Sobolev problems, outperforming existing methods.

Findings

Unconditionally stable in strong norm

Spatial fourth-order accuracy

Second-order convergence in time

Abstract

This paper proposes an unconditionally stable numerical method for solving a nonlinear Sobolev model with distributed delay. The proposed computational approach approximates the time derivative by interpolation technique whereas the spatial derivatives are approximated using the finite element approximation. This combination is simple and easy to implement. Both stability and error estimates of the constructed method are deeply analyzed in a strong norm which is equivalent to the $H^{1}$-norm. The theoretical results indicate that the constructed approach is unconditionally stable, spatial fourth-order accurate, second-order convergent in time and more efficient than a large class of numerical methods discussed in the literature for solving a general class of delay Sobolev problems. Some numerical examples are carried out to confirm the theory and demonstrate the applicability and validity of the developed technique.